Fractional parts of Dedekind sums in function fields
type: Item
instance of: scholarly article
title: Fractional parts of Dedekind sums in function fields
zbMATH Open document ID: 1365.11043
author: Yoshinori Hamahata
published in: Monatshefte für Mathematik
publication date: 2016-08-01T00:00:00Z
review text: The paper under review considers Dedekind sums in function fields, and shows some of the properties of the classical Dedekind sums also hold. Let \({\mathbb F}_q\) be the finite field with \(q\) elements, \(A={\mathbb F}_q[T]\) with \(T\) an indeterminate and \(K={\mathbb F}_q(T)\). Also let \(K_{\infty}\) be the completion of \(K\) at \(\infty=(1/T)\) and \(C_{\infty}\) be the completion of an algebraic closure of \(K_{\infty}\). For a rank \(r\) \(A\)-lattice \(\Lambda\), i.e., a finitely generated \(A\)-submodule \(\Lambda\) of rank \(r\) in \(C_{\infty}\) that is discrete in the topology of \(C_{\infty}\), there exist a Drinfeld \(A\)-module \(\phi^{\Lambda}\) of rank \(r\) over \(C_{\infty}\) and a surjective analytic function \(e_{\Lambda}\) \(:C_{\infty}\rightarrow C_{\infty}\) such that \[ e_{\Lambda}(az)=\phi_a^{\Lambda}(e_{\Lambda}(z))\;\;\text{for all}\;\; a\in A\; \] with some other properties. The Dedekind sum associated with \(\Lambda\) is defined for \(a,\,c\in A\setminus \{0\}\) with \((a,c)=1\) by \[ s_{\Lambda}(a,c)=\frac{1}{c}\sum_{0\neq \lambda\in \Lambda/c\Lambda}e_{\Lambda} \left(\frac{\lambda}{c}\right)^{-1}e_{\Lambda}\left(\frac{a\lambda}{c}\right)^{-1}\;. \] This is an analogue of the classical Dedekind sum, and the reciprocity law has been obtained for more general sums in [\textit{A. Bayad} and the author, Acta Arith. 152, No. 1, 71--80 (2012; Zbl 1301.11047)]. For a rank 1 \(A\)-lattice \(L\) which corresponds to the Carlitz module and for \(q=3\) or 2, the normalized Dedekind sum is defined as \[ {\mathcal S}(a,c)=\begin{cases}(T^3-T)s_{L}(a,c)\qquad &\text{if}\;\; q=3\,,\\ (T^4+T^2)s_{L}(a,c)\quad &\text{if}\;\;q=2\;. \end{cases} \] In this paper, the sums \({\mathcal S}(a,c)\) are investigated, and there are three main results: {\parindent=0.7cm \begin{itemize} \item[(i)] Fractional parts of the \({\mathcal S}(a,c)\) are determined~(Theorem 3.1). Also it is shown that any element of \(K\) can be a fractional part of \({\mathcal S}(a,c)…
reviewed by: Kaori Ota
Mathematics Subject Classification ID: 11F20
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11G09
zbMATH DE Number: 6608237
zbMATH Keywords: Dedekind sum
zbMATH Keywords: function field
zbMATH Keywords: Rademacher function
zbMATH Keywords: Drinfeld module
MaRDI profile type: Publication
full work available at URL: s00605-015-0781-0
OpenAlex ID: W1037389069
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DOI: 10.1007/S00605-015-0781-0
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description: scientific article; zbMATH DE number 6608237